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The Königsberg Bridge Problem

The Königsberg Bridge Problem. Leonhard “my name rhymes with boiler” Euler (1707-1783).

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The Königsberg Bridge Problem

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  1. The Königsberg Bridge Problem

  2. Leonhard “my name rhymes with boiler” Euler (1707-1783)

  3. “The problem, which I understand is quite well known, is stated as follows:  In the town of Könisberg in Prussia there is an island called Kneiphof, with two branches of the river Pregel flowing around it.  There are 7 bridges - a, b, c, d, e, f, and g - crossing the two branches.  The question is whether a person can plan a walk in such a way that he will cross each of these bridges once but not more than once.  I was told that while some denied the possibility of doing this and others were in doubt, no one maintained that it was actually possible.  On the basis of the above I formulated the following very general problem.....”

  4. a b c d e g f

  5. a b c d g f e

  6. R A B L

  7. R a b c A d B e g f L

  8. R A B L

  9. R a A

  10. R A

  11. R A B L

  12. R R R A

  13. R A B L

  14. R R R A A A B B A A B L L L

  15. R R R B A B B L L L

  16. R R R B A B B L L L

  17. R B A B B L L L

  18. R B A B B L L L

  19. R A B L L L

  20. R A B L L L

  21. R A B L

  22. R A B L

  23. The resulting figure is called a graph. The dots are its vertices and the lines are its edges. Leonard Euler was able to solve the Königsberg Bridge Problem by first modeling it with this graph.

  24. R A B L

  25. Can we trace all the edges in Euler’s graph without retracing any? If yes, how? If not, why not?

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